Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Let $a, \lambda, \mu \in \mathbb{R}$ consider the system of linear equations $ax + 2y = \lambda$, $3x - 2y = \mu$. Which of the following statement(s) is(are) correct?
If $a = -3$ then the system has infinitely many solutions for all values of $\lambda$ and $\mu$
If $a \neq -3$, then the system has a unique solution for all values of $\lambda$ and $\mu$
If $\lambda + \mu = 0$ the system has infinitely many solutions for $a = -3$
If $\lambda + \mu = 0$, then the system has no solutions for $a = -3$
Step-by-Step Solution
Key Concept: The consistency of a system of linear equations depends on comparing the determinant of the coefficient matrix Δ = -2a - 6 with the rank condition. When Δ = 0 (a = -3), the system has infinitely many solutions only if the augmented matrix ranks are equal (consistency condition λ + μ = 0), otherwise it has no solution.
For the system $ax + 2y = \lambda$ and $3x - 2y = \mu$, the coefficient matrix has determinant $\Delta = \begin{vmatrix} a & 2 \\ 3 & -2 \end{vmatrix} = -2a - 6$. When $a \neq -3$, then $\Delta \neq 0$ and the system has a unique solution. When $a = -3$, we have $\Delta = 0$, and the system is consistent only when $\lambda + \mu = 0$ (option B is correct). Option D shows that if $\lambda + \mu \neq 0$ when $a = -3$, the system becomes inconsistent.
Correct Answer: 2,3,4